Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Quick answer: A parabola is the set of points equally far from a focus and a directrix. An ellipse has distances to two foci that add to a constant, and a hyperbola has distances to two foci that differ by a constant. Completing the square rewrites any of them in standard form.
Standards: G-GPE.A.1, G-GPE.A.2, G-GPE.A.3 · Level: Grade 12 Precalculus
What is a conic section, and how do the circle and parabola work?
A conic section is a curve made by slicing a double cone with a plane. The four main shapes are the circle, parabola, ellipse and hyperbola. A circle with center (h, k) and radius r has equation (x − h)² + (y − k)² = r².
A parabola is the set of points that are the same distance from a fixed point, the focus, and a fixed line, the directrix. With vertex (h, k) and focal distance p, the upward or downward form is (x − h)² = 4p(y − k), with focus (h, k + p) and directrix y = k − p. Replace the roles of x and y for a sideways parabola.
How are ellipses and hyperbolas defined by their foci?
An ellipse is the set of points whose distances to two foci add to a constant 2a. A hyperbola is the set of points whose distances to two foci differ by a constant 2a. In standard form centered at (h, k) with a horizontal main axis:
| Conic | Equation | Foci relation | Extras |
|---|---|---|---|
| Ellipse | (x − h)²a² + (y − k)²b² = 1 | c² = a² − b² | vertices (h ± a, k) |
| Hyperbola | (x − h)²a² − (y − k)²b² = 1 | c² = a² + b² | asymptotes y − k = ±ba(x − h) |
Using c² = a² + b² for an ellipse. In an ellipse, a is the largest of the three and c is smaller than a.
How do you identify a conic by completing the square?
- Group the x terms and the y terms, and move the constant to the right.
- Factor out any leading coefficients, then complete each square (adding the same amount to both sides).
- Divide to make the right side 1. Two squares with plus signs give an ellipse (or circle); a minus sign gives a hyperbola; a single square gives a parabola.
For 4x² + 9y² − 16x + 18y − 11 = 0, write 4(x − 2)² + 9(y + 1)² = 11 + 16 + 9 = 36, so (x − 2)²9 + (y + 1)²4 = 1. It is an ellipse with center (2, −1), a = 3, b = 2 and c = √5.
Find c last. First read a and b from the denominators, then use the right relationship to locate the foci.
- Circle: (x − h)² + (y − k)² = r²; complete the square to find the center and radius.
- Parabola (x − h)² = 4p(y − k): focus (h, k + p), directrix y = k − p.
- Ellipse: c² = a² − b², and the sum of focal distances is 2a.
- Hyperbola: c² = a² + b², the difference of focal distances is 2a, and the asymptotes have slope ±b/a.
- A plus sign between squares means ellipse, a minus sign means hyperbola, one square means parabola.
Parabolas, Ellipses and Hyperbolas in Standard Form: frequently asked questions
What is the focus of a parabola?
The focus is the fixed point that defines the parabola together with the directrix. Every point on the curve is exactly as far from the focus as from the directrix. Signals arriving parallel to the axis reflect to the focus, which is why dishes use this shape.
How do I know whether a conic is an ellipse or a hyperbola?
Compare the signs of the squared terms after completing the square. Same signs give an ellipse (or a circle if the denominators match), and opposite signs give a hyperbola.
Which denominator is a² in an ellipse?
The larger denominator is a², and it sits under the variable along the major axis. If the larger denominator is under x, the major axis is horizontal; if under y, it is vertical.
What do the asymptotes of a hyperbola do?
They are the lines the branches approach but never touch. For a horizontal hyperbola centered at (h, k), they pass through the center with slopes ±b/a. Drawing the box with sides 2a and 2b locates them.
Why do I need to add to both sides when completing the square?
Adding a number to only one side changes the equation. Whatever is added to complete a square on the left, multiplied by any factored-out coefficient, must also be added to the right.
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